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Quantum Mechanics: The Other Great Revolution of the 20th Century – Part I Michael Bass, Professor Emeritus CREOL, The College of Optics and Photonics University of Central Florida © M. Bass Revolution Since Copernicus the word revolution in a scientific context means a significant change in how we think. The 20th century saw two of these: Relativity Quantum Mechanics The time frame for both was remarkably short – 1900 - 1930 © M. Bass The players in Part I Joseph Stephan, Ludwig Boltzmann, Wilhelm Wein, Lord Rayleigh, James Jeans, Max Planck, Johann Jacob Balmer, J.R. Rydberg, J. J. Thompson, Ernest Rutherford, Neils Bohr © M. Bass Problems The nagging facts that classical physics could not explain: the black body spectrum, the line spectra of atoms, the fact that atoms were stable, and others including the specific heat of solids at low temperatures. © M. Bass Spectroscopy primer Before going further it is useful to learn a little spectroscopy so that we share the same vocabulary. Spectroscopy is the study of the spectral properties of emission or absorption of various materials (e.g., gases, liquids, solids, plasmas) In the 19th century spectroscopy was done with prism (and maybe grating) dispersive elements and photographic emulsions on glass plates. light source thin slit dispersing prism lens to focus the slit onto plane of emulsion emulsion on glass plate dark room to develop the emulsion developed plate showing images of the slit (lines) and maybe some black body spectra too © M. Bass The black body spectrum Measure the radiated power of the emission from some hot object such as a hot filament at various temperatures and, as Joseph Stephan showed experimentally in 1879 and Ludwig Boltzmann showed theoretically using thermodynamic arguments, you will find that it depends on the fourth power of the filament’s temperature, e.g., P(T) = sT4. The Stephan-Boltzmann law: radiated power is proportional to T4. In 1894 Wilhelm Wein determined that the wavelength of the maximum black body radiation times the temperature of the black body was a constant, e.g. lT=2898 mm K. Unfortunately, at very long wavelengths (low temperatures) the experimental data disagreed with this result!!! © M. Bass The Rayleigh-Jeans Law Rayleigh in 1900 and with Jeans corrections in 1905 formulated the Rayleigh-Jeans Law in which the energy density of black body radiation per unit wavelength interval is proportional to the temperature divided by the fourth power of the wavelength. This worked well in the infrared but when re-written in terms of frequency it suggests that the energy density increases without bound as the frequency increases. Well, we knew that black bodies do not radiate immense amounts of high frequency light and so faced the ultraviolet catastrophe – failure of the model. © M. Bass Enter Max Karl Ernst Ludwig Planck (1858-1947) In 1900 Max Planck proposed an empirical explanation of the black body spectrum. (He did it because it worked.) Ever since Maxwell’s brilliant theory, everyone accepted the idea that electromagnetic radiation was a form of wave phenomenon. After all such waves: Interfered as waves, Diffracted as waves, Propagated as waves with speed c Were polarized as transverse waves Looked, sounded and probably smelled like waves. © M. Bass But maybe they weren’t waves Planck started a revolution by proposing that in fact they weren’t waves but particles or quanta of energy with energy proportional to the frequency that the wave would have. E = hn Notice that on the first day of the revolution we were already confronted with the waveparticle duality that we still suffer with. We consider waves to have quantum like nature and particles, quanta, to have wavelike natures when we need to. © M. Bass Assuming light is quantized Planck derived his now famous black body radiation spectrum. By fitting this curve to the data he determined the constants a and b to be 2 a = 2pc h where 5 b / lT h = 6.626 x 10-34 J s, and b =hc/ k where k= Boltzmann’s constant = 1.381 x 10-23 J/K Note that he did this fitting without the aid of a computer and he got away with two fitting parameters and one equation. He differentiated it to find the peak wavelength and he fit the derivative to Wein’s law to have a second equation to fit. He presented this to the Berlin Physical Society on October 19, 1900 – the day on which the revolution began. Rl a l e 1 1 © M. Bass If light came in quanta Could this account for the line spectra of atoms? Line spectra had been a problem for a long time. Look at the sun’s spectrum with a spectrometer and you find a continuum broken by missing lines (later recognized as absorption features of the solar gases). Look at the emission of a hydrogen discharge lamp and you see discrete spectral lines in the output. © M. Bass What was available? In the time before the planetary model of atoms, all that physicists had was the oscillating dipole. Remember this was before the electron was recognized as a particle and before the nucleus was considered to be made up of protons and neutrons (as late as 1930 we didn’t even know about neutrons). If we get a little ahead of ourselves and consider the electron-on-a-spring-attached-to-a-stationarynucleus model in the classical sense we can not account for the lines. In fact, we can not even account for stable atoms. © M. Bass A classical electron on a spring atom (no lines and unstable) We know from electromagnetics that accelerating charges radiate energy. We know that an electron on a spring goes through accelerated motion and so must radiate energy. If it does so it must lose energy and eventually crash into the nucleus. Even some sort of specific, characteristic modes of the oscillating electron would decay away and so the spectrum before the crash should be continuous. In classical physics atoms were not stable and spectra were continuous. Clearly there was a problem. - + © M. Bass The experiments and the data First you had to prepare atoms of H, N,… not molecules of H2, N2 … Second you had to perform precision spectroscopy in the days when photography was barely understood, emulsions were not sensitive, emulsion spectral responses were not known, you did not have CCDs or even Polaroid film, you had to develop your (home made) plates in the dark without distorting the emulsion, then, if you hadn’t yet damaged the emulsion, you had to measure the distances between the observed spectral lines with high precision without damaging the plates, and finally, if you trusted your calibrations, you could report the wavelengths (frequencies) that you found. © M. Bass Balmer In 1885, Johann Jacob Balmer, a school teacher in Basel, Switzerland published a paper in the Proceedings of the Scientific Society of Basel that was to shake physics in a very fundamental way. Balmer was not a physicist. He was a numerologist. He liked to play with numbers. He noticed that the wavelengths of the four lines reported by Angstrom in 1868 could be represented in terms of “a basic number, h=364.50682 nm, as (9/5)h, (4/3)h,(25/21)h and (9/8)h.” Then he recognized that these fractions were 9/5, 16/12, 25/21 and 36/32. 2 2 2 2 The numerators were 3 , 4 , 5 , and 6 , and 2 2 2 2 2 2 2 2 the denominators were 3 -2 , 4 -2 , 5 -2 and 6 -2 . 2 m The wavelengths were related by the formula lh 2 2 m 2 He even predicted other lines in the series and that this series of lines would end when m = 3 at 656.2 nm. © M. Bass Implications Somehow the H atom was emitting light with wavelengths related by integers. With guidance from a numerologist’s skill with numbers, physicists went out and determined many other regularities in spectra and line sequences. It was clear that integers mattered. In 1890, J. R. Rydberg claimed he had been using Balmer’s formula long before it had been published. This might have been “sour grapes” but we will never really know. Rydberg’s work was much more thorough than Balmer’s and greatly expanded the utility of the idea that integers mattered in spectra. © M. Bass The formulae worked but why? The situation was similar to classical mechanics when there was Copernicus’ model, Brahe’s data, Kepler’s formulae, and Galileo’s concepts. The synthesis was missing!! Then the next big clue came from J. J. Thompson and others. The discovery of the electron The wrong model of the atom Still the electron would spiral in as it radiated a continuum of light and atoms would not be stable. Either something was wrong with the scattering experiments that pointed to this model or something was wrong with classical physics. © M. Bass Bohr’s experience In 1912, after about half a year working in J. J. Thompson’s lab at Cambridge, Neils Bohr was gently booted out. He was booted because he disagreed with Thompson’s model for the atom. The plum pudding model He went to work for Ernest Rutherford (a New Zealander) (the planetary model) at Manchester. This is the first step in the synthesis that would lead to Quantum Mechanics as we know it. Bohr would establish his model for the H atom and it would lead to the general acceptance of Rutherford’s model of the atom. © M. Bass Everything is quantized Where Planck introduced the concept of quanta to electromagnetic energy, Bohr introduced the concept of quanta to the motion of matter. Today we say very loosely that Bohr “quantized matter”. We already had conceived of quanta of charge and mass so the general idea was not too hard to accept. The problem was, “Why should something so odd be so intrinsically a part of nature?” © M. Bass Bohr’s model (a zeroth order picture of how the atom would work) We could picture it but we didn’t know why it worked as he said. Just as in Newton’s day people didn’t know what forces were but they could conceive what they did. Bohr stated the obvious – atoms have E = hn stable states and when the state of the atom changes from one of these states to another a quantum of electromagnetic energy is released (or absorbed) having energy equal to the energy difference between the two states. This demanded that the states have energies described by integers. E = En E = hn E = En-1 hn = En – En-1 © M. Bass Quantization as a tool In the process of quantizing the atom Bohr showed how to avoid the singularity of the electron spiraling in and reaching hyper-relativistic speeds and the infinity of density that the classical planetary model predicted. Bohr gave modern theorists the tool of quantization to remove singularities. © M. Bass Previews of Coming Attractions Next time in Quantum Mechanics, Part II we will: Go into the planetary model problem. Look at Bohr’s model in more depth. See that Bohr’s derivation of the Rydberg constant was crucial to the acceptance of Quantum Mechanics. Examine the Correspondence Principle. Discuss quantized angular momentum and spin. Introduce de Broglie’s waves. Preview Part III © M. Bass